Valence-band anticrossing in mismatched III-V semiconductor alloysK. Alberi,1,2J. Wu,1,2W. Walukiewicz,1K. M. Yu,1O. D. Dubon,1,2S. P. Watkins,3C. X. Wang,3X. Liu,4Y.-J. Cho,4andJ. Furdyna41Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA2Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA3Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S64Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA共Received 26 May 2006; revised manuscript received 15 November 2006; published 16 January 2007兲We show that the band gap bowing trends observed in III-V alloys containing dilute concentrations of Sb orBi can be explained within the framework of the valence-band anticrossing model. Hybridization of theextended p-like states comprising the valence band of the host semiconductor with the close-lying localizedp-like states of Sb or Bi leads to a nonlinear shift of the valence-band edge and a reduction of the band gap.The two alloys GaSbxAs1−xand GaBixAs1−xare explored in detail, and the results are extrapolated to additionalsystems.DOI: 10.1103/PhysRevB.75.045203 PACS number共s兲: 71.20.Nr, 71.55.Eq, 78.30.FsI. INTRODUCTIONAmid the current activity in semiconductor alloy research,materials containing either Bi or Sb have emerged as poten-tial candidates for long wavelength applications. Comparableto III-N-V compounds, recent reports show that the band gapof GaAs redshifts considerably upon the addition of only afew atomic percent of Bi or Sb.1–9Furthermore, it has beensuggested that GaBixAs1−xexhibits other anomalous proper-ties, such as a reduced temperature dependence of the bandgap as well as giant spin-orbit splitting.5,10However, little ispresently known about the exact mechanisms by which thesealloying elements influence the properties of III-V com-pounds. Using GaSbxAs1−xand GaBixAs1−xas exemplarysystems, we propose a theoretical model of the electronicstructure of III-Sb-V and III-Bi-V alloys based upon thatdescribing other similar dilute semiconductor alloy systems.The band anticrossing 共BAC兲 model was developed toexplain the unusual features of the electronic structure of aclass of semiconductors termed highly mismatched alloys共HMAs兲 and was initially applied to those systems in whichhighly electronegative isoelectronic impurity atoms are in-corporated onto the anion sublattice of a III-V or II-VI com-pound semiconductor.11Well-known examples includeGaNxAs1−x共Ref. 12兲 and ZnOxTe1−x共Ref. 13兲 in which alarge-scale bowing of the band gap,14an increase of the elec-tron effective mass,15and a reduction of the pressure depen-dence of the band gap11have been observed upon the addi-tion of N or O, respectively. In contrast to alloys with mixedcation elements, highly electronegative elements on the an-ion sublattice introduce localized defect states sufficientlyclose to the conduction-band edge of the host matrix to un-dergo a quantum anticrossing interaction with the extendedstates of the matrix.16This interaction produces a splitting ofthe conduction band into E−and E+levels, with the down-ward movement of the former leading to the band gap bow-ing observed in these dilute alloys. The term “mismatched”may also refer to a disparity in the atomic size of the alloyingelement occupying the anion sublattice. The defect states oflarge-sized impurities with first ionization energies less thanthat of the host anion often lie near the valence-band edge ofthe host semiconductor. Hybridization between the two re-sults in a modification of the valence band in a similar man-ner to the restructuring of the conduction band. As the atomicdiameters of Sb and Bi are the largest of the group V ele-ments, we expect such interactions to occur in GaSbxAs1−xand GaBixAs1−x, and these alloys may be considered in thecontext of the valence-band anticrossing 共VBAC兲 model. In-deed, the explicit restructuring of the valence band that hasbeen experimentally observed in other III-V and II-VI alloysystems, including GaN1−xAsx, ZnSe1−xTex, and ZnSxTe1−x,has been attributed to such anticrossing interactions.17,18In this paper we consider the effects of the substitution ofAs atoms by Sb and Bi to form dilute GaSbxAs1−xandGaBixAs1−x, respectively, and show that the band gap bowingand increase of the spin-orbit splitting observed in these al-loys are well explained by the valence-band anticrossingmodel 共VBAC兲.II. VALENCE-BAND ANTICROSSING MODELThe valence-band structures of As-rich GaSbxAs1−xandGaBixAs1−xalloys were developed within the realm of thek·p formalism. In each case, the standard 6 ⫻ 6 Hamiltoniandescribing the four ⌫8and two ⌫7valence bands of GaAswas modified by the addition of the six localized p-like statesof the minority Sb or Bi atoms with basis functions identicalto those describing the host semiconductor.19To address theissue of the random placement of impurity atoms in the crys-tal lattice we have used a previously developed method ofconfigurational averaging carried out within the coherent po-tential approximation, which has been shown to restoretranslational symmetry.11,20The resulting 12⫻12 matrixHamiltonian is given asPHYSICAL REVIEW B 75, 045203 共2007兲1098-0121/2007/75共4兲/045203共6兲 ©2007 The American Physical Society045203-1HV=冨冨冨冨冨H␣␤0i␣冑2− i冑2␤V共x兲000 0 0␣*L0␤iD冑2i冑32␣0V共x兲00 0 0␤*0L−␣− i冑32␣*iD冑200V共x兲00 00␤*−␣*H− i冑2␤*− i␣*冑2000V共x兲00− i␣*冑2− iD冑2i冑32␣i冑2␤S00000V共x兲0i冑2␤*− i冑32␣*− iD冑2i␣冑20S0000 0V共x兲V共x兲00000Eimp000 0 00V共x兲00 0 00Eimp000 000V共x兲00 000Eimp00000 0V共x兲0 0 000Eimp0000 00V共x兲0 0000Eimp-so000 000V共x兲0000 0Eimp-so冨冨冨冨冨. 共1兲The individual parameters in Eqs. 共2a兲–共2f兲 are those ofthe original 6 ⫻ 6 valence-band matrix modified by the addi-tion of the terms ⌬EVBMx to H and L and ⌬ESOx to S, whichtake into account the linear change in the valence-band maxi-mum and spin-orbit split-off band energies, respectively, be-tween the two endpoint compounds as described by the vir-tual crystal approximation 共VCA兲. Here ⌬EVBMand ⌬ESOare the total differences in valence-band maximum and spin-orbit split-off band energies between the endpoint com-pounds,H